There is no way to link Oyster cards together so you can’t add the two-together railcard discount to Oyster. You may find that contatless is better for you as you don’t need to pay for the card or faff about with topping up. You would each need a separate contactless card or device (smart phone/watch etc). This will be fine for travel around central London because the daily cap of £7.40 each is cheaper than any ticket you could get with your railcard.
On the day you go to Hampton Court it depends what else you do. If it’s just a return from Waterloo then use contactless and it’s £4.30 single each way as long as you don’t start before 0930. Total £8.60 each. If you intend to do anything else that day, even just a bus ride, then get two off-peak zones 1-6 travelcards with your railcard at £9.10 each, then use the travelcard for everything that day. Hampton Court station is in zone 6.
Hope that helps.